Finite Group Databasebeta

← #28401355

Subgroups — #28401355

Order 14464.

420 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20299109C211yesmaximal, characteristic
2#20298936C212yesmaximal, complemented
3#20298928C211yesmaximal, characteristic, complemented
4#20298911C211yesmaximal, characteristic, complemented
5#20298837C212yesmaximal, complemented
6#20298829C212yesmaximal, complemented
7#20298758C212yesmaximal, complemented
8#20298754C211yesmaximal, characteristic, complemented
9#20298737C211yesmaximal, characteristic, complemented
10C2×C22⋊#11744751C211yesmaximal, characteristic, complemented
11C2×C22⋊Q8×C113C211yesmaximal, characteristic, complemented
12C2×#14268647C2211yescharacteristic
13C2×#14268637C2212yescomplemented
14C2×#14268634C2212yescomplemented
15C22×#11744751C2212yes—
16#16813605—21no—
17#16813601C2211yescharacteristic
18#16813601C2211yescharacteristic
19#16813596C2211yescharacteristic
20#16813596C2211yescharacteristic
21#16813591—21no—
22C4×C22⋊D113—21no—
23C4×C22⋊D113—21no—
24#16813581C2211yescharacteristic
25#16813555C2212yescomplemented
26#16813553C2212yescomplemented
27#16813546C2212yescomplemented
28#16813546C2212yescomplemented
29#16813545C2212yescomplemented
30#16813542C2212yescomplemented
31C22⋊#11744751C2211yescharacteristic
32C22⋊#11744751C2212yes—
33C22⋊#11744751C2211yescharacteristic
34C22×Q8×C113C2211yescharacteristic
35C23×C4×C113C2211yescharacteristic
36C22⋊Q8×C113—21no—
37C22⋊Q8×C113—21no—
38C22⋊Q8×C113C2212yes—
39C22⋊Q8×C113C2212yes—
40C2×C4⋊C4×C113—21no—
41C2×C4⋊C4×C113C2211yescharacteristic
42C2×C22⋊C4×C113—21no—
43#14268647—42no—
44C2×Dic226C2311yescharacteristic
45C2×D452C2311yescharacteristic
46C2×C22⋊D113—21no—
47C2×C22⋊D113—21no—
48C2×C22⋊D113—21no—
49C2×C4×D113—21no—
50#14268637—42no—
51#14268634—42no—
52C2×#11744751—42no—
53C2×#11744751C2312yes—
54C2×#11744751C2312yes—
55C4⋊Dic113C2311yescharacteristic
56C4⋊Dic113C2311yescharacteristic
57#14268626—21no—
58#14268626—21no—
59C22⋊Dic113—21no—
60C22⋊Dic113—21no—
61C22⋊Dic113—21no—
62#14268624—21no—
63#14268624—21no—
64C4×Dic113—21no—
65C24×C113D411yescharacteristic
66C2×Q8×C113—21no—
67C2×Q8×C113—42no—
68C2×Q8×C113C2312yes—
69C22×C4×C113—21no—
70C22×C4×C113—21no—
71C22×C4×C113—42no—
72C22×C4×C113D411yescharacteristic
73C22×C4×C113D411yescharacteristic
74C22×C4×C113D411yescharacteristic
75C22×C4×C113C2311yescharacteristic
76C22×C4×C113C2311yescharacteristic
77C22×C4×C113C2311yescharacteristic
78C4⋊C4×C113—22no—
79C4⋊C4×C113—22no—
80C4⋊C4×C113—21no—
81C4⋊C4×C113C2312yes—
82C22⋊C4×C113—22no—
83C22⋊C4×C113—22no—
84C23×C113—22no—
85C23×C113—22no—
86C23×C113—41no—
87C23×C113C2×D411yescharacteristic
88C23×C113C2×D411yescharacteristic
89C23×C113C2×D411yescharacteristic
90C22×D113—21no—
91Q8×C113—22no—
92Q8×C113—22no—
93Q8×C113—42no—
94C2×C4×C113—22no—
95C2×C4×C113—22no—
96C2×C4×C113—22no—
97C2×C4×C113—22no—
98C2×C4×C113—22no—
99C2×C4×C113—41no—
100C2×C4×C113—41no—
101C2×C4×C113—41no—
102C2×C4×C113—42no—
103C2×C4×C113—42no—
104C2×C4×C113C2×D411yescharacteristic
105C2×C4×C113C2×D411yescharacteristic
106C2×C4×C113C2×D411yescharacteristic
107C2×C4×C113C2411yescharacteristic
108C22⋊D113—42no—
109C22⋊D113—42no—
110C22⋊D113—42no—
111C2×Dic113—21no—
112C2×Dic113—21no—
113C2×Dic113—21no—
114D452—22no—
115C4×D113—42no—
116Dic226—22no—
117#11744751—22no—
118#11744751—22no—
119C22×C113—21no—
120C22×C113—21no—
121C22×C113—21no—
122C22×C113—21no—
123C22×C113—21no—
124C22×C113—21no—
125C22×C113—21no—
126C22×C113—44no—
127C22×C113#13512yes—
128C22×C113#13712yes—
129C22×C113C22×D411yescharacteristic
130C2×D113—41no—
131C2×D113—41no—
132C2×D113—41no—
133C4×C113—21no—
134C4×C113—21no—
135C4×C113—21no—
136C4×C113—42no—
137C4×C113—42no—
138C4×C113H(5,𝔽2)12yes—
139Dic113—41no—
140Dic113—41no—
141Dic113—41no—
142C2×C113—22no—
143C2×C113—22no—
144C2×C113—41no—
145C2×C113#53411yescharacteristic
146C2×C113C2×#13511yescharacteristic
147C2×C113C2×#13711yescharacteristic
148D113—81no—
149#3190—1131nomaximal, Sylow
150C113#319011yescharacteristic, complemented, Sylow
151#525—1131no—
152C2×C22⋊Q8D11311yescharacteristic, complemented
153#485—1132no—
154#483—1131no—
155#480—1131no—
156#467—1132no—
157#465—1132no—
158#452—1132no—
159#451—1131no—
160#448—1131no—
161C2×C22⋊C8—1131no—
162C2×#41—1131no—
163C22×Q8C2×D11311yescharacteristic
164C23×C4C2×D11311yescharacteristic
165C2×Q16—1132no—
166C2×SD16—1132no—
167C22×C8—1132no—
168#123—2261no—
169C22⋊Q8—21no—
170C22⋊Q8—21no—
171C22⋊Q8—1131no—
172C22⋊Q8—1131no—
173C22⋊Q8C2×D11312yes—
174C22⋊Q8C2×D11312yes—
175#121—1131no—
176#121—1131no—
177#120—2261no—
178C4×D4—2261no—
179C4×D4—2261no—
180#117—1131no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4C2×D11311yescharacteristic
183C2×C22⋊C4—21no—
184#107—1132no—
185#106—1132no—
186#103—1132no—
187#103—1132no—
188#103—1132no—
189#102—1132no—
190C22⋊C8—1131no—
191C22⋊C8—1131no—
192C22⋊C8—1132no—
193C24C22⋊D11311yescharacteristic
194#41—4522no—
195C2×Q8—21no—
196C2×Q8—42no—
197C2×Q8—1131no—
198C2×Q8C22×D11312yes—
199C2×D4—1131no—
200C2×D4—2261no—
201C2×D4—2261no—
202C2×D4—2261no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—42no—
206C22×C4—2261no—
207C22×C4C22⋊D11311yescharacteristic
208C22×C4C22⋊D11311yescharacteristic
209C22×C4C22⋊D11311yescharacteristic
210C22×C4C22×D11311yescharacteristic
211C22×C4C22×D11311yescharacteristic
212C22×C4C22×D11311yescharacteristic
213Q16—4522no—
214SD16—4522no—
215C2×C8—1132no—
216C2×C8—1132no—
217C2×C8—4522no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—21no—
221C4⋊C4—1131no—
222C4⋊C4—1131no—
223C4⋊C4—2261no—
224C4⋊C4—2261no—
225C4⋊C4C22×D11312yes—
226C22⋊C4—22no—
227C22⋊C4—22no—
228C22⋊C4—2261no—
229C22⋊C4—2261no—
230C22⋊C4—2261no—
231C22⋊C4—2261no—
232C22⋊C4—2261no—
233C42—2261no—
234C23—22no—
235C23—22no—
236C23—41no—
237C23—2261no—
238C23C2×C22⋊D11311yescharacteristic
239C23C2×C22⋊D11311yescharacteristic
240C23C2×C22⋊D11311yescharacteristic
241Q8—22no—
242Q8—22no—
243Q8—42no—
244Q8—2262no—
245D4—4522no—
246D4—4522no—
247D4—4522no—
248D4—2262no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—4522no—
260C2×C4—2261no—
261C2×C4—2261no—
262C2×C4—2261no—
263C2×C4C2×C22⋊D11311yescharacteristic
264C2×C4C2×C22⋊D11311yescharacteristic
265C2×C4C2×C22⋊D11311yescharacteristic
266C2×C4C23×D11311yescharacteristic
267C8—2262no—
268C8—2262no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—44no—
277C22—4521no—
278C22—4521no—
279C22—4521no—
280C22#1681364212yes—
281C22#1681365312yes—
282C22C22×C22⋊D11311yescharacteristic
283C4—21no—
284C4—21no—
285C4—21no—
286C4—42no—
287C4—42no—
288C4—4521no—
289C4—4521no—
290C4—4521no—
291C4#1681366712yes—
292C2—22no—
293C2—22no—
294C2—41no—
295C2—9041no—
296C2#2029915711yescharacteristic
297C2C2×#1681364211yescharacteristic
298C2C2×#1681365311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
FittingC2×C22⋊Q8×C113C2
socleC22×C113C22×D4
prime radical#20299109C2
hypercenterC2×C22⋊Q8D113
hypocenterC113#3190
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #20298936C2semidirect
3. #20298928C2semidirect
4. #20298911C2semidirect
5. #20298837C2semidirect
6. #20298829C2semidirect
7. #20298758C2semidirect
8. #20298754C2semidirect
9. #20298737C2semidirect
10. C2×C22⋊#11744751C2semidirect
11. C2×C22⋊Q8×C113C2semidirect
13. C2×#14268637C22semidirect
14. C2×#14268634C22semidirect
25. #16813555C22semidirect
26. #16813553C22semidirect
27. #16813546C22semidirect
28. #16813546C22semidirect
29. #16813545C22semidirect
30. #16813542C22semidirect
150. C113#3190semidirect
152. C2×C22⋊Q8D113semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202991092
#202989362
#202989282
#202989112
#202988372
#202988292
#202987582
#202987542
#202987372
C2×C22⋊#117447512
C2×C22⋊Q8×C1132
#3190113